Bring 4 cups water and broth to a boil in a 4-qt. saucepan. Stir in grits, salt, and pepper. Bring to a boil, and cook, uncovered, 8 minutes or until very thick. Add cheese, stirring until it melts. Remove from heat, and pour grits into a 13" x 9" pan coated with cooking spray. Set aside to cool completely.
While grits cool, process green onions and cilantro in a small food processor until minced. With processor running, slowly add oil, and process until well blended. Pour cilantro oil into a small saucepan; set aside.
Cut cooled grits into 8 rectangles. Cut each rectangle into 2 triangles. Place a 12" cast-iron skillet over medium-high heat until hot; coat skillet with cooking spray. Add half of grits rectangles to skillet, taking care to keep rectangles together for easier turning. Cook 3 minutes; coat tops with cooking spray; turn and cook on other side 3 minutes or until golden. Transfer grits to a serving platter, separating rectangles into triangles, and keep warm. Repeat procedure with remaining grits rectangles.
Heat oil mixture over medium heat just until warm. Remove from heat, and stir in vinegar. Drizzle 1 tsp. oil evenly over each triangle; serve with remaining oil, if desired.
Make Ahead: Cut cooled grits as directed above. Cover and store in refrigerator. Brown grits and prepare herb oil just before serving.